\magnification=1200
\hsize=4in
\overfullrule=0pt
\input amssym
%\def\frac#1 #2 {{#1\over #2}}
\def\emph#1{{\it #1}}
\def\em{\it}
\nopagenumbers
\noindent
%
%
{\bf Andrei Zelevinsky}
%
%
\medskip
\noindent
%
%
{\bf Semicanonical Basis Generators of the Cluster Algebra of Type $A_1^{(1)}$}
%
%
\vskip 5mm
\noindent
%
%
%
%
We study the cluster variables and ``imaginary" elements of the
semicanonical basis for the coefficient-free cluster algebra of affine
type $A_1^{(1)}$. A closed formula for the Laurent expansions of
these elements was given by P.Caldero and the author. As a
by-product, there was given a combinatorial interpretation of the
Laurent polynomials in question, equivalent to the one obtained by
G.Musiker and J.Propp. The original argument by P.Caldero and the
author used a geometric interpretation of the Laurent polynomials due
to P.Caldero and F.Chapoton. This note provides a quick,
self-contained and completely elementary alternative proof of the same
results.
\bye